Post by Al Moritz on May 11, 2009 10:15:00 GMT
See Humphrey's article:
bedejournal.blogspot.com/2009/01/why-is-universe-so-big.html
I have always wondered about an important aspect of this issue, and this article, satisfying as it is in many ways, does not really answer the question. Neither does anything else that I have read, including Stephen Barr's book Modern Physics and Ancient Faith and his article mentioned below. Here is a discussion that I recently had with Stephen Barr on the issue by email. Barr's answers are technical, but worth reading: they describe the reason why the universe has to expand so fast, there is no other choice.
(AM = Al Moritz, SB = Stephen Barr)
AM:
I have a question is about the expansion rate of the universe.
In your article:
www.firstthings.com/article.php3?id_article=2208
you said (and I have seen the same argument made by others):
"Physics can also suggest why the universe has to be so large. The laws of gravity discovered by Einstein relate the size of the universe directly to its age. The fact that the universe is many billions of light-years across is related to the fact that it has lasted several billions of years. Perhaps we would be less daunted by a cozy little universe the size, say, of a continent. But such a universe would have lasted only a few milliseconds. Even a universe the size of the solar system would have lasted only a few hours. A universe constructed in such a way as to evolve life may well have had to extend widely in space as well as in time. It may well be that the frightening expanses that are so often said to be a sign of human insignificance may actually, like so many other features of our strange universe, point to man, as they also proclaim the glory of God."
Why would that be? Could not the universe have expanded at a slower rate than close to the speed of light at great distances from us *) or even greater than the speed of light in the unobservable part of the universe)? If so, could it not have been much smaller while still being billions of years old -- let's say with much less matter so that it does not collapse onto itself at that lower rate of expansion? What requires the high speed of expansion after the Big Bang? Simple particle physics -- if so, why? Or is such huge expansion rate required by Einstein's equations?
*) The galaxy Hydra at 3.5 billion light years distance from us apparently moves away from us at 38,000 miles/sec, as measured by redshift
SB:
As to the relation of age to size of the universe: If one assumes the universe is homogeneous and isotropic, then one gets from Einstein's Equations the so-called Friedmann Equations. Given a particular "equation of state" for the matter that fills the universe (e.g. radiation, "dust", cosmological constant, etc, or some combination of them) then the Friedmann Equations tell you the rate of change of the "scale factor" as a function of the scale factor itself. So one does not have the freedom to make the universe expand as slowly as one likes.
AM:
thanks for the reply. I do not quite understand the following:
Given a particular "equation of state" for the matter that fills the universe (e.g. radiation, 'dust", cosmological constant, etc, or some combination of them) then the Friedmann Equations tell you the rate of change of the "scale factor" as a function of the scale factor itself.
Is radiation related to size too? What do you mean by the rate of change of the "scale factor" as a function of the scale factor itself?
SB:
There are three quantities that enter the Friedman Equations (in the simple case we are talking about of a homogeneous isotropic universe):
the "scale factor" (usually denoted by "a"), the density of matter (denoted by the Greek letter rho), and the pressure of the matter (denoted by p). These are all functions of time t.
If the universe is spatially "closed", then it has a finite volume and radius. The scale factor could be taken to be the radius. In other words "a" is just how big the universe is. If the universe is spatially "open", then it is infinite in size. In an open universe, one can take "a" to be the size of some finite portion of the universe.
The "equation of state", tells you the properties of the matter, and is a relationship between the density and pressure (rho and p). For example, for "radiation" (i.e. matter whose particles are going close to the speed of light, such as a photons or neutrinos) the equation of state is that p = rho/3. For "dust" (meaning any matter made up of particles going very slowly compared to the speed of light (such as hydrogen gas, clouds of dust, stars, etc.) the equation of state is that p = 0 (approximately). The "cosmological constant" can be thought of as a fluid with p = - rho. Once you use the equation of state to relate rho and p, you is down to two variables, say "a" and rho. Then there is an equation that expresses the fact that energy is conserved. That will relate rho to "a". For example, for radiation rho is proportional to "a" to the minus fourth power. This leaves just one unknown function of time, say "a", and one equation, called the Friedman equation, which follows from Einstein's field equations of gravity. The Friedman equation gives a relation between the first time derivative of "a" and "a" itself. In other words, if you specify "a", then the Friedman equation tells you the rate at which "a" is changing. That is why the rate of expansion is tied to the size of the universe.
AM:
Thanks for your detailed reply. I am trying to understand this:
This leaves just one unknown function of time, say "a", and one equation, called the Friedman equation, which follows from Einstein's field equations of gravity. The Friedman equation gives a relation between the first time derivative of "a" and "a" itself. In other words, if you specify "a", then the Friedman equation tells you the rate at which "a" is changing. That is why the rate of expansion is tied to the size of the universe.
But if "a" is the size of the universe, or some finite portion of the universe, would not then the rate of expansion change with its size? If the universe was very small right after the Big Bang, would not the rate of expansion have been much smaller than what it is now? Or does this have to do with the "first time derivative", which may change with time, so that a young universe would still expand at a great rate even though it was small?
SB:
The fact that I said that "the rate of expansion is tied to the size of the universe" makes you ask "would not the rate of expansion have been much smaller than what it is now?" The answer is no. The rate of expansion may have been BIGGER also -- in fact it was. To say that two things are "related" or "tied" together does not mean that they have a direct relation, they may also have an inverse relation. In this particular case they do. If the universe is filled mostly with radiation ("radiation dominated") as it was in its very early stages, then the rate of expansion goes as time to the -1/2 power; if the universe is "matter dominated", as it was for most of its history up till now, then the rate of expansion goes as time to the -1/3 power -- in both cases, the universe's expansion slows with time. For the last few billion years, the universe has been dominated by "dark energy" (i.e. "cosmological constant"). That has made the expansion speed up.
AM:
Thank you for your further explanations. I think the issue now becomes clearer to me. So I suppose in the end the expansion of the universe is, even though not exclusively, also related to simple particle physics after all, given the difference between "radiation dominated" and "matter dominated" situations (determining the "pressure").
SB:
You're welcome.
bedejournal.blogspot.com/2009/01/why-is-universe-so-big.html
I have always wondered about an important aspect of this issue, and this article, satisfying as it is in many ways, does not really answer the question. Neither does anything else that I have read, including Stephen Barr's book Modern Physics and Ancient Faith and his article mentioned below. Here is a discussion that I recently had with Stephen Barr on the issue by email. Barr's answers are technical, but worth reading: they describe the reason why the universe has to expand so fast, there is no other choice.
(AM = Al Moritz, SB = Stephen Barr)
AM:
I have a question is about the expansion rate of the universe.
In your article:
www.firstthings.com/article.php3?id_article=2208
you said (and I have seen the same argument made by others):
"Physics can also suggest why the universe has to be so large. The laws of gravity discovered by Einstein relate the size of the universe directly to its age. The fact that the universe is many billions of light-years across is related to the fact that it has lasted several billions of years. Perhaps we would be less daunted by a cozy little universe the size, say, of a continent. But such a universe would have lasted only a few milliseconds. Even a universe the size of the solar system would have lasted only a few hours. A universe constructed in such a way as to evolve life may well have had to extend widely in space as well as in time. It may well be that the frightening expanses that are so often said to be a sign of human insignificance may actually, like so many other features of our strange universe, point to man, as they also proclaim the glory of God."
Why would that be? Could not the universe have expanded at a slower rate than close to the speed of light at great distances from us *) or even greater than the speed of light in the unobservable part of the universe)? If so, could it not have been much smaller while still being billions of years old -- let's say with much less matter so that it does not collapse onto itself at that lower rate of expansion? What requires the high speed of expansion after the Big Bang? Simple particle physics -- if so, why? Or is such huge expansion rate required by Einstein's equations?
*) The galaxy Hydra at 3.5 billion light years distance from us apparently moves away from us at 38,000 miles/sec, as measured by redshift
SB:
As to the relation of age to size of the universe: If one assumes the universe is homogeneous and isotropic, then one gets from Einstein's Equations the so-called Friedmann Equations. Given a particular "equation of state" for the matter that fills the universe (e.g. radiation, "dust", cosmological constant, etc, or some combination of them) then the Friedmann Equations tell you the rate of change of the "scale factor" as a function of the scale factor itself. So one does not have the freedom to make the universe expand as slowly as one likes.
AM:
thanks for the reply. I do not quite understand the following:
Given a particular "equation of state" for the matter that fills the universe (e.g. radiation, 'dust", cosmological constant, etc, or some combination of them) then the Friedmann Equations tell you the rate of change of the "scale factor" as a function of the scale factor itself.
Is radiation related to size too? What do you mean by the rate of change of the "scale factor" as a function of the scale factor itself?
SB:
There are three quantities that enter the Friedman Equations (in the simple case we are talking about of a homogeneous isotropic universe):
the "scale factor" (usually denoted by "a"), the density of matter (denoted by the Greek letter rho), and the pressure of the matter (denoted by p). These are all functions of time t.
If the universe is spatially "closed", then it has a finite volume and radius. The scale factor could be taken to be the radius. In other words "a" is just how big the universe is. If the universe is spatially "open", then it is infinite in size. In an open universe, one can take "a" to be the size of some finite portion of the universe.
The "equation of state", tells you the properties of the matter, and is a relationship between the density and pressure (rho and p). For example, for "radiation" (i.e. matter whose particles are going close to the speed of light, such as a photons or neutrinos) the equation of state is that p = rho/3. For "dust" (meaning any matter made up of particles going very slowly compared to the speed of light (such as hydrogen gas, clouds of dust, stars, etc.) the equation of state is that p = 0 (approximately). The "cosmological constant" can be thought of as a fluid with p = - rho. Once you use the equation of state to relate rho and p, you is down to two variables, say "a" and rho. Then there is an equation that expresses the fact that energy is conserved. That will relate rho to "a". For example, for radiation rho is proportional to "a" to the minus fourth power. This leaves just one unknown function of time, say "a", and one equation, called the Friedman equation, which follows from Einstein's field equations of gravity. The Friedman equation gives a relation between the first time derivative of "a" and "a" itself. In other words, if you specify "a", then the Friedman equation tells you the rate at which "a" is changing. That is why the rate of expansion is tied to the size of the universe.
AM:
Thanks for your detailed reply. I am trying to understand this:
This leaves just one unknown function of time, say "a", and one equation, called the Friedman equation, which follows from Einstein's field equations of gravity. The Friedman equation gives a relation between the first time derivative of "a" and "a" itself. In other words, if you specify "a", then the Friedman equation tells you the rate at which "a" is changing. That is why the rate of expansion is tied to the size of the universe.
But if "a" is the size of the universe, or some finite portion of the universe, would not then the rate of expansion change with its size? If the universe was very small right after the Big Bang, would not the rate of expansion have been much smaller than what it is now? Or does this have to do with the "first time derivative", which may change with time, so that a young universe would still expand at a great rate even though it was small?
SB:
The fact that I said that "the rate of expansion is tied to the size of the universe" makes you ask "would not the rate of expansion have been much smaller than what it is now?" The answer is no. The rate of expansion may have been BIGGER also -- in fact it was. To say that two things are "related" or "tied" together does not mean that they have a direct relation, they may also have an inverse relation. In this particular case they do. If the universe is filled mostly with radiation ("radiation dominated") as it was in its very early stages, then the rate of expansion goes as time to the -1/2 power; if the universe is "matter dominated", as it was for most of its history up till now, then the rate of expansion goes as time to the -1/3 power -- in both cases, the universe's expansion slows with time. For the last few billion years, the universe has been dominated by "dark energy" (i.e. "cosmological constant"). That has made the expansion speed up.
AM:
Thank you for your further explanations. I think the issue now becomes clearer to me. So I suppose in the end the expansion of the universe is, even though not exclusively, also related to simple particle physics after all, given the difference between "radiation dominated" and "matter dominated" situations (determining the "pressure").
SB:
You're welcome.
